How many types of ice exist and what is their nomenclature? I know that the ice on Earth is $I_h$, but why so?
The first thing to note about ice water is:
All ice is not crystalline and not all crystalline water is ice
$LDA$ : Low Density Amorphous
$HDA$ : High Density Amorphous
$VHDA$ : Very High Density Amorphous
All crystalline ice is tetrahedrically coordinated (where the oxygen atoms are in fixed positions relative to each other but the hydrogen atoms may or may not be disordered but obeying the 'ice rules'), and the polymorphs differ primarily in H-bond angles and next-nearest neighbour separations.
Ice I encompasses all ice found on naturally on earth (in a stable form).
This is usually split into two phases called $I_h$ and $I_c$.
Hexagonal ice (ice $I_h$) is 'ordinary ice', the form of all natural snow and ice on Earth. This is evidenced in the six-fold symmetry in ice crystals grown from water vapor (that is, snow flakes).
Cubic ice (ice $I_c$) is a solid form of water that can form in high clouds. Ice Ic is metastable, and irreversibly converts to ice $I_h$. Both forms contain similar layers of hexagonal rings, but differs in the way these layers are connected.
For more information on their structure and properties of ice, check out 'the Many Phases of Ice'.
Martin Chaplin maintains a fairly thorough and well referenced online resource on water, which includes a comprehensive section on the ice phases.
Finally, Google Books has the definitive 'the Physics of Ice' by Petrenko & Whitworth available online.
There are different types of ice, depending on its cristalline structure.
My guess would be that Ice $I_h$ and $I_c$ are named directly after their structure, which is respectively hexagonal and cubic.
The numbering, which makes Ice I, Ice II, Ice III and so on till Ice XV, seems to be arbitrary, probably ordered in sequence as they have been discovered. This also tells you about the number of different types of ice known: 16 (or 19 if we considering amorphous ice in its three forms).